540201 Statistics for Engineer Week II and Week

540201 Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution

Random Variables Controllable Variables Output Input Uncontrollable Variables Random Variable : A Numerical variable whose measured value can change from one replicate of the experiment to another

3 -2 Random Variables �Discrete random variables �Continuous random variables

3 -3 Probability �The chance of “x” �A degree of belief �A relative frequency between “event frequency” to the “outcome frequency” Histogram of Compressive Strength 0. 275 22 0. 2125 0. 175 14 0. 125 0. 075 0. 0250. 0375 17 10 0. 05 0. 025 2 3 6 4 2

3 -4 Continuous Random Variables �Cumulative Distribution Function (cdf)

Continuous Random Variables �Probability Density Function (pdf)

Continuous Random Variables �Mean and Variance

Example 3. 5

Continuous Uniform Distribution

Continuous Uniform Distribution

Continuous Uniform Distribution

Continuous Uniform Distribution

3 -5. 1 Normal Distribution (Gaussian)

Normal Distribution

Normal Distribution

Normal Distribution

Normal Distribution �Ex 3 -11 �Ex 3 -12

Normal Distribution

Normal Distribution

t-Distribution is unknown �Small sample size �Degree of freedom (k) = n-1 �Significant level = �t , k �When

t-Distribution

Exponential Distribution

Exponential Distribution

3 -7 Discrete Random Variables • Probability Mass Function (pmf)

Discrete Random Variables �Cumulative Distribution Function (cdf)

Discrete Random Variables �Mean and Variance

3 -8 Binomial Distribution �A Bernoulli Trial

Binomial Distribution

Binomial Distribution Example 3 -28 Bit transmission errors: Binomial Mean and Var

3 -9 Poison Distribution The random variable X that equals the number of events in a Poison process is a Poison random variable with parameter >0, and the probability mass function of X is The mean and variance of X are

3 -9 Poison Distribution

3 -9 Poison Distribution

3 -9 Poison Distribution

3 -10 Normal Approximation to the Binomial and Poisson Distributions �Normal Approximation to the Binomial

3 -10 Normal Approximation to the Binomial and Poisson Distributions

3 -10 Normal Approximation to the Binomial and Poisson Distributions

3 -10 Normal Approximation to the Binomial and Poisson Distributions �Normal Approximation to the Poisson

3 -10 Normal Approximation to the Binomial and Poisson Distributions

3 -10 Normal Approximation to the Binomial and Poisson Distributions

Q &A